Matrix solutions and nonsigularity

Question

I'm new to matrices and have some trouble with them, if anyone can help me with the next question I would appreciate it a lot.

Let A be a 3x3 matrix and suppose that $2a_1 + a_2 + 4a_3 = 0$

a. How many solutions will the system Ax=0 have?

b. Is A nonsingular?

Thanks in advance :)

Answer 1

Suppose $a_i$ refers to the $i$-th row of $A$.

$$2a_1+a_2+4a_3=0$$

implies that the rows are linearly dependent which implies that $A$ is singular and $Ax=0$ has infinitely many solutions.

Answer 2

It means $u = (2,1,4)^\top$ is a solution of $Ax=0$.As are all scalar multiples $\alpha u$. So we have infinite many solutions.

Non-singular matrices have only the null vector as solution of $Ax=0$, which is not the case for this matrix.